Irrational Numbers - AN2 I

Irrational Numbers - AN2 I

The Number Systems

In this course we will talk about six different number systems:

1. Natural Numbers: N

2. Whole Numbers: W

3. Integers: Z

4. Rational Numbers: Q

5. Irrational Numbers I

6. Real Numbers: R

What Is An Irrational Number?

What Is A Rational Number?

? A rational number is any number that can be

expressed as the quotient of two integers.

? An irrational number is a number that cannot

be expressed as the quotient of two integers.

? A repeating, or terminating ¡Ìinteger.

i.e. 1.56721343, or 1.12, or .49

? A non-repeating, and non-terminating

decimal.

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i.e 1.12349123639..., ¦Ð, and 2

Practice:

Identify Whether each number is Rational, or Irrational:

i.

¡Ì

34

ii.

1

2

iii.

36

2

iv.

¦Ð

36

vi.

vii.

viii.

v. 0.654

¡Ì

¡Ì

¡Ì

0.75

9

0.25

ix. 0.333

1

What Is A Radical?

? A radical is a number that involves a root

? A mixed radical is a number that is part radical, and part integer. Mixed :

? An entire radical An entire radical is a number that only has a root. The entire number is under the

radical.

Expressing An Entire Radical As A Mixed Radical In Simplest Form

In order to express

an entire radical as a mixed radical, we use prime factorization:

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Example: 48

1. Prime factorization: 48 = 2 ¡¤ 2 ¡¤ 2 ¡¤ 2 ¡¤ 3

2. Identify groups based on index (in this case 2): 48 = 22 ¡¤ 22 ¡¤ 3

¡Ì

¡Ì

3. Write with radical: 48 = 22 ¡¤ 22 ¡¤ 3

¡Ì ¡Ì ¡Ì

¡Ì

¡Ì

¡Ì ¡Ì

4. Note that a b = ab, so 48 = 22 22 3

¡Ì

¡Ì

5. Simplifying: 48 = 2 ¡¤ 2 ¡¤ 3

¡Ì

¡Ì

6. So 48 = 4 3

i.

¡Ì

32

ii.

¡Ì

3

40

Practice:

Convert The Following To Mixed Radicals In Simplest Form:

i.

ii.

¡Ì

27

¡Ì

3

iii.

80

iv.

2

¡Ì

75

¡Ì

3

128

Expressing A Mixed Radical as an Entire Radical.

¡Ì

Example: 2 5:

¡Ì ¡Ì

¡Ì ¡Ì

¡Ì

1. Begin by rewriting coefficient as as a radical: 2 5 = 22 5 = 4 5

¡Ì

¡Ì ¡Ì

¡Ì

¡Ì ¡Ì

2. Next, remember a b = ab, so 4 5 = 4 ¡¤ 5

¡Ì

¡Ì

3. Next, simplify: 4 ¡¤ 5 = 20

¡Ì

¡Ì

4. So 2 5 = 20.

¡Ì

i. 5 5

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ii. 2 7

Practice:

Convert The Following Into Entire Radicals:

¡Ì

i. 5 7

¡Ì

iv. 10 2

¡Ì

ii. 2 6

¡Ì

v. 3 3 2

¡Ì

iii. 3 13

¡Ì

vi. 2 3 5

3

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